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We relate two ways to renormalize the Brownian loop measure on the Riemannsphere. One by considering Brownian loop measure on the sphere minus a smalldisk which is known as the normalized Brownian loop measure; the other one bytaking the measure on simple loops induced by the outer boundary of theBrownian loops, known as Werner's measure. This result allows us to interpretthe Loewner energy as an Onsager--Machlup functional for SLE$_\kappa$ loopmeasure for any fixed $\kappa \in (0, 4]$, and more generally, for anyMalliavin--Kontsevich--Suhov loop measure of the same central charge.

Onsager Machlup functional for $\text{SLE}_{\kappa}$ loop measures